Causal inference problems often involve continuous treatments, such as dose, duration, or frequency. However, identifying and estimating standard dose-response estimands requires that everyone has some chance of receiving any level of the exposure (i.e., positivity). To avoid this assumption, we consider stochastic interventions based on exponentially tilting the treatment distribution by some parameter $δ$ (an incremental effect); this increases or decreases the likelihood a unit receives a given treatment level. We derive the efficient influence function and semiparametric efficiency bound for these incremental effects under continuous exposures. We then show estimation depends on the size of the tilt, as measured by $δ$. In particular, we derive new minimax lower bounds illustrating how the best possible root mean squared error scales with an effective sample size of $n / δ$, instead of $n$. Further, we establish new convergence rates and bounds on the bias of double machine learning-style estimators. Our novel analysis gives a better dependence on $δ$ compared to standard analyses by using mixed supremum and $L_2$ norms. Finally, we define a "reflected" exponential tilt around any interior point and show that taking $δ\to \infty$ yields a new estimator of the dose-response curve across the treatment support.

翻译：因果推断问题常涉及连续处理变量，如剂量、持续时间或频率。然而，识别和估计标准剂量-响应估计量要求每个个体都有机会接受任意水平的暴露（即正性假设）。为规避该假设，我们考虑基于以参数$δ$（增量效应）对处理分布进行指数倾斜的随机干预；这会增加或减少个体接受特定处理水平的可能性。我们推导了连续暴露下这些增量效应的有效影响函数与半参数效率界。随后证明估计效果取决于倾斜幅度，即$δ$的大小。特别地，我们推导了新的极小极大下界，阐明最优均方根误差如何随有效样本量$n/δ$而非$n$变化。此外，我们建立了双重机器学习类估计量的新收敛速率与偏差界。通过采用混合上确界范数与$L_2$范数，我们的创新分析相比标准方法获得了对$δ$更优的依赖关系。最后，我们定义了围绕任意内点的"反射"指数倾斜，并证明当$δ\to \infty$时可得到处理支撑集上剂量-响应曲线的新估计量。